The main theorem of this paper is a little involved (though the proof is straightforward using a well-known idea) but the immediate corollaries are interesting. For example, take a complex normed vector space A which is also a normed algebra with identity under each of two multiplications * and ∘. Then these multiplications coincide if and only if there exists α such that ‖a ∘ b ‖ ≦ α ‖ a * b ‖ for a, b in A. This is a condition for the two Arens multiplications on the second dual of a Banach algebra to be identical. By taking * to be the multiplication of a Banach algebra and ∘ to be its opposite, we obtain the condition for commutativity given in (3). Other applications are concerned with conditions under which a bilinear mapping between two algebras is a homomorphism, when an element lies in the centre of an algebra, and a one-dimensional subspace of an algebra is a right ideal. An example shows that the theorem is false for algebras over the real field, but Theorem 2 gives the parallel result in this case.

Multiresolution is investigated on the basis of shift-invariant spaces. Given a finitely generated shift-invariant subspace S of L2(ℝd), let Sk be the 2k-dilate of S (k∈ℤ). A necessary and sufficient condition is given for the sequence {Sk}k∈ℤ to fom a multiresolution of L2(ℝd). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skew-symmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.

The following paper contains little that can be regarded as new mathematical information. It aims only at showing, or rather at emphasising, the correspondence which exists between two geometrical theories which are related to each other in the same way as the arithmetical theories of multiplication and division. Such value, therefore, as it possesses is primarily pedagogical.

In this note all small latin letters denote rational integers. We write k ≧ 1, s ≧ 1 and consider the simultaneous equations

A solution of these equations is said to be non-trivial if no set {xiu} is a permutation of another set {xiv}. In 1851 Prouhet constructed a non-trivial solution of these equations with j = sk and Lehmer has recently found a parametric solution for the same j. Here I give two alternative elementary proofs of Lehmer's result. Lehmer's own proof depends on the ideas of generating functions, exponentials, differentiation, matrices, and complex roots of unity, though all at a fairly simple level. One of my proofs requires only the factor theorem for a polynomial and the other only the multinomial theorem for a positive integral index.

Let k be a field of characteristic p>0. We classify all finite p-groups G satisfying the inequality p−2|G|≦t(G) < p−1|G|, where t(G) is the nilpotency index of the Jacobson radical of k[G].

When studying the solutions of elliptic boundary value problems in a bounded, smoothly bounded domain D⊂Rn we often encounter the formula

where u(x)∈C2(D)∩C′() is a solution of the second order self-adjoint elliptic equation

and denotes differentiation along the inward normal to ∂D at x∈∂D.

In a recent paper (6) the present author has shown that, for an element a of a Banach algebra A, the condition

for all x∈A and some constant α is equivalent to [x, a]∈Rad a for all x∈A; it turns out that α may be replaced by |α|σ It is the purpose of the present note to investigate a related condition

In this paper the laws of addition and subtraction of vectors were considered, and examples of their extreme usefulness in geometrical applications were given.

For any sequence (aj) of complex numbers and for any ρ > ½, we construct an entire function F with the following properties. F has order ρ, mean type, each aj is a deficient value of F, and F is given by F(z)=f(g(z)), where f and g are transcendental entire functions. This complements a result of Goldstein. We also construct, for any ρ>½, an entire function G of order p, mean type, such that liminf,→ ∞ T(r, G)/T(r, G′)>1.

Several papers on the subject of spatial distance in General Relativity appeared a few years ago, and a simple extension of this idea to any pair of points in any Riemannian space was given by me in a thesis. A distance invariant was defined, and this was found to depend upon a certain two-point invariant which was first introduced by H. S. Ruse in a study of Laplace's Equation. This invariant, now written ρ and defined in (3), has lately re-appeared, and it may now be of interest to publish the results found earlier. These include a geometrical interpretation of ρ, a simple method of calculation, and an expansion as a power series in the geodesic arc. The dependence of ρ upon the geodesic arc is also considered.

In my laboratory we make great use of the method of tracing lines of force described in Glazebrook and Shaw's Practical Physics. The field is modified in various ways; for instance, the fixed magnets are sometimes placed so as to somewhat resemble in their disposition the field magnets of a dynamo: sometimes a circular piece of soft iron is placed in the field, and the effect of its induced magnetization examined. The students like these exercises, and the results (of which some are exhibited) are very beautiful. We use steel bars about 8 cm. long for fixed magnets, and small rather heavy lozenge-shaped needles, about one to two cm. long, as what I shall call pointers.

It is known that the behaviour of an electromagnetic field is consistent with it possessing a linear momentum density (1, p. 103)

(Girogi Units are used and the system of notation is the usual one).